Problem 41 - Algebra 2 and Trigonometry

Question

a. Write each fraction in simplest radical form. b. Use a calculator to find a rational approximation for the given fraction. c. Use a calculator to find a rational approximation for the fraction in simplest form. \(\frac{4}{\sqrt{3}-1}\)

Step-by-Step Solution

Answer

The fraction simplifies to \( 2\sqrt{3} + 2 \) and approximates to 5.464.

Step 1: Rationalize the Denominator

To simplify the expression \( \frac{4}{\sqrt{3} - 1} \), multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of \( \sqrt{3} - 1 \) is \( \sqrt{3} + 1 \). Thus, we perform the following multiplication: \[ \frac{4}{\sqrt{3} - 1} \times \frac{\sqrt{3} + 1}{\sqrt{3} + 1} = \frac{4(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} \] The denominator simplifies using the difference of squares: \[ (\sqrt{3} - 1)(\sqrt{3} + 1) = 3 - 1 = 2 \] So the expression becomes: \[ \frac{4(\sqrt{3} + 1)}{2} = \frac{4\sqrt{3} + 4}{2} = 2\sqrt{3} + 2 \].

Step 2: Rational Approximation of Original Fraction

Use a calculator to find a decimal approximation for the original fraction \( \frac{4}{\sqrt{3} - 1} \). Calculate \( \sqrt{3} \) to get approximately 1.732, and then find the value of the original fraction. The value is approximately \( 4.464 \).

Step 3: Rational Approximation of Simplified Form

Convert the simplest radical form \( 2\sqrt{3} + 2 \) into a decimal approximation. Calculate \( 2\sqrt{3} \) using \( \sqrt{3} \approx 1.732 \). \[ 2\sqrt{3} + 2 \approx 2(1.732) + 2 = 3.464 + 2 = 5.464 \].

Key Concepts

Simplest Radical Form

The simplest radical form is where a fraction or expression involving a radical is written in the most simplified version possible. This often requires rationalizing the denominator, which means eliminating any radical expressions in the denominator. For instance, given the fraction \( \frac{4}{\sqrt{3} - 1} \), the denominator contains the radical \( \sqrt{3} \).

• The goal is to rewrite this expression in such a way that the radical is not present in the denominator. This simplifies further calculations and makes analysis easier.
• After rationalizing, in this case, the expression becomes \( 2\sqrt{3} + 2 \), which is the simplest radical form.

Understanding how to find the simplest radical form can greatly ease computations and enhance clarity in mathematical problems.

Conjugate

The conjugate of a two-term radical expression arises when we change the sign between the terms. When dealing with expressions like \( \sqrt{3} - 1 \), its conjugate is \( \sqrt{3} + 1 \). Utilizing conjugates plays a crucial role in rationalizing denominators.

• By multiplying the numerator and the denominator of a fraction by the conjugate of the denominator, the resulting expression eliminates the radical from the denominator.
• This technique simplifies the fraction significantly, as seen with \( \frac{4}{\sqrt{3} - 1} \), transforming it to \( \frac{4\sqrt{3} + 4}{2} \) when simplified.

Conjugates are extremely useful, particularly when simplifying expressions to subsequently evaluate or further manipulate them.

Rational Approximation

Rational approximation involves converting irrational numbers into a rational form, usually a decimal, to better comprehend or solve a problem. This is typically done using calculators.

• For our initial expression \( \frac{4}{\sqrt{3} - 1} \), finding a rational approximation helps compare the efficiency of the original and simplified forms.
• Calculating \( \sqrt{3} \approx 1.732 \) provides the basis. Thus for \( \frac{4}{1.732 - 1} \), the rational approximation is approximately \( 4.464 \).

This practical method turns abstract irrational numbers into understandable figures, aiding in various real-world and theoretical applications.

Difference of Squares

The difference of squares is a useful algebraic identity which simplifies expressions involving conjugates. The identity is encapsulated in the formula: \( (a-b)(a+b) = a^2 - b^2 \). This concept is particularly important when rationalizing denominators in radical expressions.

• In practice, it simplifies the expression \( (\sqrt{3} - 1)(\sqrt{3} + 1) \) to \( 3 - 1 = 2 \).
• Applying this identity effectively reduces the complexity of radical fractions, transforming them into a more manageable form like \( \frac{4(\sqrt{3} + 1)}{2} \).

Understanding the difference of squares facilitates the elimination of radicals in expressions, making subsequent calculations and manipulations more straightforward.